Example of a function of two variables that cannot be an $R$-function
We note that the definition of R-functions depends on the choice of a certain surjection and pose the problem of the construction of a function of two variables that is not an R-function for any choice of a surjective mapping. It is shown that the function $x_1 x_2 − 1$ possesses this property. We p...
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2010
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860508851549241344 |
|---|---|
| author | Velichko, I. G. Stegantseva, P. G. Величко, И. Г. Стеганцева, П. Г. Величко, И. Г. Стеганцева, П. Г. |
| author_facet | Velichko, I. G. Stegantseva, P. G. Величко, И. Г. Стеганцева, П. Г. Величко, И. Г. Стеганцева, П. Г. |
| author_sort | Velichko, I. G. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:39:03Z |
| description | We note that the definition of R-functions depends on the choice of a certain surjection and pose the problem of the construction of a function of two variables that is not an R-function for any choice of a surjective mapping. It is shown that the function $x_1 x_2 − 1$ possesses this property. We prove a theorem according to which, in the case of finite sets, every mapping is an $R$-mapping for a proper choice of a surjection. |
| first_indexed | 2026-03-24T02:31:47Z |
| format | Article |
| fulltext |
UDK 517.51+510.644
Y. H. Velyçko, P. H. Stehanceva
(ZaporoΩ. nac. un-t)
PRYMER FUNKCYY DVUX PEREMENNÁX,
KOTORAQ NE MOÛET BÁT| R - FUNKCYEJ
We call attention to the fact that the definition of an R-function depends on the choice of some
surjection. The problem of the construction of a function of two variables, that is not the R-function at
any choice of the surjective mapping, is formulated. We show that the function x x1 2 1− possesses this
property and prove the theorem, according to which, in the case of finite sets, any mapping should be the
R-mapping under the appropriate choice of the surjection.
Zverneno uvahu na te, wo oznaçennq R - funkci] zaleΩyt\ vid vyboru deqko] sgr’[kci]. Sfor-
mul\ovano zadaçu pro pobudovu tako] funkci] dvox zminnyx, qka ne [ R - funkci[g ni pry qkomu
vybori sgr’[ktyvnoho vidobraΩennq. Pokazano, wo funkciq x x1 2 1− ma[ taku vlastyvist\.
Dovedeno teoremu pro te, wo u vypadku skinçennyx mnoΩyn bud\-qke vidobraΩennq bude R - vi-
dobraΩennqm pry slußnomu vybori sgr’[kci].
Vvedenye. V poslednee vremq v razlyçn¥x pryloΩenyqx ßyroko prymenqgtsq
R - funkcyy, vvedenn¥e V. L. Rvaçev¥m v 1963 h. [1]. Ony pozvolqgt dostatoçno
prosto reßyt\ obratnug zadaçu analytyçeskoj heometryy — zapysat\ uravne-
nye kryvoj, kotoraq ohranyçyvaet zadannug oblast\. ∏to okaz¥vaetsq polez-
n¥m pry reßenyy zadaç matematyçeskoho prohrammyrovanyq, optymal\noho ras-
poloΩenyq obæektov na ploskosty, teoryy ustojçyvosty, matematyçeskoj fy-
zyky y dr.
Naybolee polno teoryq R - funkcyj yzloΩena v fundamental\n¥x monohra-
fyqx avtora metoda [2 – 4]. Otmetym takΩe rabot¥ [5 – 8], v kotor¥x reßagt-
sq zadaçy mexanyky y matematyçeskoho modelyrovanyq.
Poçty vse rabot¥ ob R - funkcyqx posvqwen¥ pryloΩenyqm. Matematyçes-
koe obosnovanye πtoho apparata ymeetsq tol\ko v publykacyqx avtora metoda.
Vmeste s tem yzuçenye samoho ponqtyq R - funkcyy qvlqetsq neobxodym¥m dlq
obobwenyq rezul\tatov na nov¥e obæekt¥.
V lyterature pryvodqtsq opredelenye R - funkcyy y prymer¥ R - funkcyj,
no net ny odnoho prymera funkcyy, kotoraq ne qvlqetsq R - funkcyej. V dan-
noj rabote pryveden prymer konkretnoj funkcyy y dokazano, çto ona ne moΩet
b¥t\ R - funkcyej.
Postanovka zadaçy. Napomnym opredelenye R - otobraΩenyq y R - funkcyy
[2, c. 101]). Pust\ zadan¥ mnoΩestvo X , soderΩawee ne menee k > 1 πlemen-
tov, y sgræektyvnoe otobraΩenye Sk : X → Bk , hde Bk = { 0, 1, … , k – 1 } .
Opredelenye. OtobraΩenye f : X
n → X
m
naz¥vaetsq R - otobraΩenyem,
esly suwestvuet takaq funkcyq k - znaçnoj lohyky F B Bk
n
k
m: → , kotoraq
vmeste s f obrazuet kommutatyvnug dyahrammu
X X
B B
n f m
k
n F
k
m
S Sk
n
k
m
→
↓ ↓
→
yly, druhymy slovamy, esly
© Y. H. VELYÇKO, P. H. STEHANCEVA, 2010
270 ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 2
PRYMER FUNKCYY DVUX PEREMENNÁX, KOTORAQ … 271
S fk
m � = F Sk
n� . (1)
Zdes\ S X Bk
n n
k
n: → , S x x xk
n
n( , , , )1 2 … = ( )( ), ( ), , ( )S x S x S xk k k n1 2 … .
Pry πtom funkcyq F k - znaçnoj lohyky naz¥vaetsq soprovoΩdagwej dlq f.
Esly X = R , to R - otobraΩenye naz¥vaetsq R - funkcyej.
Prymer. Pust\ X = R . Zadadym sgræektyvnoe otobraΩenye
S t3( ) =
0 0
1 0
2 0
, ,
, ,
, .
esly
esly
esly
t
t
t
<
=
>
PokaΩem, çto funkcyq f x x( , )1 2 = x x1 2 qvlqetsq R - funkcyej.
Poskol\ku f : R
2 → R
1
, to n = 2, m = 1. Funkcyg F zadadym sledugwym
obrazom:
X
X
F
1
2
0 0 0 1 1 1 2 2 2
0 1 2 0 1 2 0 1 2
2 1 0 1 1 1 0 1 2
Dlq dokazatel\stva toho, çto funkcyq f qvlqetsq R - funkcyej, nuΩno
vzqt\ vsevozmoΩn¥e znaçenyq x1, x2 y proveryt\ v¥polnenye ravenstva (1).
Esly, naprymer,
x1 < 0, x2 = 0,
to
f x x( , )1 2 = x x1 2 = 0, S f x x3 1 2( )( , ) = 1.
S druhoj storon¥,
S x x3
2
1 2( , ) = ( )( ), ( )S x S x3 1 3 2 = ( , )0 1 , F S x x( )( , )3
2
1 2 = F( , )0 1 = 1.
Znaçyt, dlq πtoho sluçaq ravenstvo (1) v¥polnqetsq:
S f x x3 1 2( )( , ) = 1 = F S x x( )( , )3
2
1 2 .
Ostal\n¥e 8 sluçaev (kaΩdaq yz peremenn¥x otrycatel\na, ravna nulg
yly poloΩytel\na) proverqgtsq analohyçno.
Pryvedennoe v¥ße opredelenye qvlqetsq, na samom dele, opredelenyem R -
otobraΩenyq otnosytel\no zadannoj sgræekcyy Sk , kotoraq, v svog oçered\,
opredelqet y çyslo k .
Lehko pokazat\, çto funkcyq g x x( , )1 2 = x x1 2 1− uΩe ne budet R - funk-
cyej otnosytel\no zadannoho otobraΩenyq S3 .
Voz\mem x1 = 1, x2 = t > 0. Tohda
S x x3
2
1 2( , ) = ( )( ), ( )S x S x3 1 3 2 = ( )( ), ( )S S t3 31 = ( , )2 2 .
S druhoj storon¥,
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 2
272 Y. H. VELYÇKO, P. H. STEHANCEVA
S g x x3 1 2( )( , ) = S t3 1 1( )⋅ − = S t3 1( )− .
Znaçyt, v sylu (1), dolΩno v¥polnqt\sq ravenstvo F( , )2 2 = S t3 1( )− .
Poskol\ku levaq çast\ ne zavysyt ot t , pravaq toΩe ne dolΩna zavyset\ ot t .
No πto neverno:
S
t3 0 5= ,
= S3 0 5 1( , )− = S3 0 5( , )− = 0 ≠ 2 = S3 1( ) = S3 2 1( )− = S
t3 2= ,
a znaçyt, funkcyq g x x( , )1 2 = x x1 2 1− uΩe ne budet R - funkcyej otnosytel\-
no zadannoho otobraΩenyq.
Ne ysklgçen tot fakt, çto ona budet R - funkcyej otnosytel\no druhoho
otobraΩenyq Sk .
Cel\g dannoj rabot¥ qvlqetsq postroenye prymera funkcyy dvux peremen-
n¥x, kotoraq ne qvlqetsq R - funkcyej ny dlq kakoho sgræektyvnoho otobra-
Ωenyq.
Osnovnaq çast\. DokaΩem, çto funkcyq g x x( , )1 2 = x x1 2 1− ne qvlqetsq
R - funkcyej.
Esly zadannaq funkcyq g : R
2 → R
1
est\ R - funkcyq, to dolΩn¥ suwest-
vovat\ natural\noe k > 1 y sgræektyvnaq funkcyq S : R
1 → B k takye, çto
v¥polnqetsq ravenstvo
F S x S x( )( ), ( )1 2 = S x x( )1 2 1− ∀ ∈x x R1 2, . (2)
Na mnoΩestve R
1
vvedem bynarnoe otnoßenye ∼ sledugwym obrazom:
x ∼ y ⇔ S x( ) = S y( ) .
Lehko ubedyt\sq, çto ono qvlqetsq otnoßenyem πkvyvalentnosty. Esly
x y1 1∼ ∧ x y2 2∼ , to
S x x( )1 2 1− = F S x S x( )( ), ( )1 2 = F S y S y( )( ), ( )1 2 = S y y( )1 2 1− .
Takym obrazom, esly
g x x( , )1 2 = x x1 2 1−
qvlqetsq R - funkcyej, to
x y1 1∼ ∧ x y2 2∼ ⇒ ( ) ( )x x y y1 2 1 21 1− −∼ . (3)
Esly x2 = y2 = t, to x y2 2∼ , y sootnoßenye (3) prynymaet vyd
x y1 1∼ ⇒ ( ) ( )x t y t1 11 1− −∼ ∀ ∈t R . (4)
Suwestvugt dva vozmoΩn¥x varyanta stroenyq klassa πlementov, πkvyvalent-
n¥x nulg. Rassmotrym yx.
1. Suwestvuet nenulevoe çyslo x , πkvyvalentnoe nulg. V¥berem x1 = 0,
y1 = x , y podstavym v (4). V rezul\tate poluçym
tx − 1 ∼ – 1 ∀ ∈t R . (5)
PoloΩyv v (5) t = 1/x , budem ymet\
0 ∼ – 1. (6)
Poskol\ku S — sgræekcyq y k ≥ 2, suwestvuet πlement
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 2
PRYMER FUNKCYY DVUX PEREMENNÁX, KOTORAQ … 273
b � 0. (7)
PoloΩyv v (5) t = ( )/b x+ 1 , poluçym b ∼ −1. Takym obrazom,
b ∼ ∼−1 0 ⇒ b ∼ 0 ,
çto protyvoreçyt (7). Sledovatel\no, πtot varyant ne moΩet b¥t\ realyzovan.
2. Ne suwestvuet nenulev¥x çysel, πkvyvalentn¥x nulg, t. e. klass πkvy-
valentnosty, soderΩawyj nul\, sostoyt yz odnoho πlementa.
Poskol\ku kolyçestvo klassov πkvyvalentnosty ravno k , t. e. qvlqetsq ko-
neçn¥m (v pryncype, moΩno b¥lo b¥ rassmatryvat\ y sçetnoe mnoΩestvo klas-
sov), suwestvuet klass, soderΩawyj, po krajnej mere, dva razlyçn¥x nenule-
v¥x πlementa a y b.
PoloΩyv v (4) t = 1/a , x1 = a , y1 = b , poluçym
a b∼ ⇒ a
a
b
a
⋅ −
⋅ −
1
1
1
1∼ ⇒ 0 1∼
b
a
− ⇒
b
a
− =1 0 ⇒ b a= .
Poluçennoe protyvoreçye (po uslovyg πlement¥ a y b razlyçn¥) pokaz¥vaet,
çto y πtot varyant ne moΩet realyzov¥vat\sq.
Takym obrazom, sootnoßenye (4) ne moΩet v¥polnqt\sq ny pry kakom v¥bore
çysla k ≥ 2 y ny pry kakom sposobe zadanyq otobraΩenyq S . Sledovatel\no,
utverΩdenye o tom, çto funkcyq g x x( , )1 2 = x x1 2 1− ne moΩet b¥t\ R - funk-
cyej, dokazano.
Sluçaj koneçn¥x mnoΩestv. Otmetym, çto dlq lgboho koneçnoho mnoΩe-
stva postroenye takoho prymera nevozmoΩno v sylu sledugwej teorem¥.
Teorema. Dlq lgboho koneçnoho mnoΩestva X vsehda moΩno podobrat\
takug sgræekcyg Sk , otnosytel\no kotoroj zadannoe otobraΩenye f : X
n →
→ X
m
budet R - otobraΩenyem.
Dokazatel\stvo. PoloΩym k = X y v¥berem v kaçestve Sk kakug-ny-
bud\ byekcyg meΩdu mnoΩestvamy X y Bk . OtobraΩenye f zapyßem çerez
koordynatn¥e otobraΩenyq:
f x( ) = ( )( ), ( ), , ( )f x f x f xm1 2 … ,
hde
x = ( , , , )x x xn1 2 … , x Xi ∈ .
Yskomaq funkcyq F k - znaçnoj lohyky odnoznaçno opredelqetsq svoymy
koordynatn¥my funkcyqmy Fi :
F b( ) = ( )( ), ( ), , ( )F b F b F bm1 2 … ,
hde
b = ( , , , )b b bn1 2 … , b Bi k∈ .
Ravenstvo (1) πkvyvalentno seryy toΩdestv
F S x S x S xi k k k n( )( ), ( ), , ( )1 2 … = S f x x xk i n( )( , , , )1 2 … , i = 1, … , m . (8)
Opredelym funkcyy Fi sledugwym obrazom:
F b b bi n( , , , )1 2 … = S f S b S b S bk i k k k n( ( ))( ), ( ), , ( )− − −…1
1
1
2
1 , i = 1, … , m . (9)
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 2
274 Y. H. VELYÇKO, P. H. STEHANCEVA
Poskol\ku S qvlqetsq byekcyej, pravaq çast\ (9) opredelqetsq odnoznaçno
dlq vsex znaçenyj lohyçeskyx peremenn¥x, y, sledovatel\no, funkcyy Fi op-
redelen¥ korrektno. V¥brann¥e funkcyy udovletvorqgt ravenstvam (8), a zna-
çyt, yskom¥e sgræekcyq Sk y funkcyq lohyky F najden¥.
Teorema dokazana.
Zaklgçenye. V dannoj rabote postroen konkretn¥j prymer funkcyy, ko-
toraq ne moΩet b¥t\ R - funkcyej ny pry kakom v¥bore sgræekcyy. Predlo-
Ωenn¥j metod moΩno lehko obobwyt\ dlq dokazatel\stva toho fakta, çto
funkcyy vyda ax x x bn1 2 … + pry n > 1 y ab ≠ 0 ne mohut b¥t\ R - funkcyq-
my. V dal\nejßem planyruetsq poluçenye klassov funkcyj, πlement¥ koto-
r¥x ne mohut b¥t\ R - funkcyqmy. Predstavlqet ynteres y poluçenye obweho
kryteryq toho, qvlqetsq ly zadannaq funkcyq R - funkcyej.
1. Rvaçev V. L. Ob analytyçeskom opysanyy nekotor¥x heometryçeskyx obæektov // Dokl. AN
USSR. – 1963. – 153, #T4. – S.T765 – 767.
2. Rvaçev V. L. Teoryq R - funkcyj y nekotor¥e ee pryloΩenyq. – Kyev: Nauk. dumka, 1982. –
552 s.
3. Rvaçev V. L., Synekop N. S. Metod R - funkcyj v zadaçax teoryy upruhosty y plastyçnosty.
– Kyev: Nauk. dumka, 1990. – 216 s.
4. Kravçenko V. F., Rvaçev V. L. Alhebra lohyky, atomarn¥e funkcyy y vejvlet¥ v fyzyçes-
kyx pryloΩenyqx. – M.: Fyzmatlyt, 2006. – 416 s.
5. Kravçenko V. F., Gryn A. V. Prymenenye teoryy R - funkcyj y vejvletov k reßenyg krae-
v¥x zadaç πllyptyçeskoho typa // ∏lektromahnytn¥e voln¥ y πlektronn¥e system¥. – 2009.
– # 3. – S.T4 – 39.
6. Slesarenko A. P., Safonov N. A. R - funkcyy, varyacyonno-strukturn¥j y yteracyonn¥e
metod¥ v ydentyfykacyy y matematyçeskom modelyrovanyy nelynejn¥x processov
teploprovodnosty v oblastqx s ystoçnykamy πnerhyy // Dop. NAN Ukra]ny. – 2007. – # 1. –
S.T94 – 99.
7. Kurpa L., Pigun G., Amabili M. Nonlinear vibrations of shallow shells with complex boundary: R -
functions method and experimental // J. Sound and Vibration. – 2007. – 306, Issues 3-5. –
P. 580 – 600.
8. Mathieu Hursin, Shanjie Xiao, Tatjana Jevremovic. Synergism of the method of characteristic, R -
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P. 1116 – 1133.
Poluçeno 03.07.09,
posle dorabotky — 19.11.09
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 2
|
| id | umjimathkievua-article-2863 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | rus English |
| last_indexed | 2026-03-24T02:31:47Z |
| publishDate | 2010 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/79/561b54508754d5160c5c0a3e06022779.pdf |
| spelling | umjimathkievua-article-28632020-03-18T19:39:03Z Example of a function of two variables that cannot be an $R$-function Пример функции двух переменных, которая не может быть $R$-функцией Velichko, I. G. Stegantseva, P. G. Величко, И. Г. Стеганцева, П. Г. Величко, И. Г. Стеганцева, П. Г. We note that the definition of R-functions depends on the choice of a certain surjection and pose the problem of the construction of a function of two variables that is not an R-function for any choice of a surjective mapping. It is shown that the function $x_1 x_2 − 1$ possesses this property. We prove a theorem according to which, in the case of finite sets, every mapping is an $R$-mapping for a proper choice of a surjection. Звернено увагу на те, що означення $R$-функції залежить від вибору деякої сюр'єкції. Сформульовано задачу про побудову такої функції двох змінних, яка не є $R$-функцією ні при якому виборі сюр'єктивиого відображення. Показано, що функція $x_1 x_2 − 1$ має таку властивість. Доведено теорему про те, що у випадку скінченних множин будь-яке відображення буде $R$- відображенням при слушному виборі сюр'єкції. Institute of Mathematics, NAS of Ukraine 2010-02-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2863 Ukrains’kyi Matematychnyi Zhurnal; Vol. 62 No. 2 (2010); 270–274 Український математичний журнал; Том 62 № 2 (2010); 270–274 1027-3190 rus en https://umj.imath.kiev.ua/index.php/umj/article/view/2863/2478 https://umj.imath.kiev.ua/index.php/umj/article/view/2863/2479 Copyright (c) 2010 Velichko I. G.; Stegantseva P. G. |
| spellingShingle | Velichko, I. G. Stegantseva, P. G. Величко, И. Г. Стеганцева, П. Г. Величко, И. Г. Стеганцева, П. Г. Example of a function of two variables that cannot be an $R$-function |
| title | Example of a function of two variables that cannot be an $R$-function |
| title_alt | Пример функции двух переменных, которая не может быть $R$-функцией |
| title_full | Example of a function of two variables that cannot be an $R$-function |
| title_fullStr | Example of a function of two variables that cannot be an $R$-function |
| title_full_unstemmed | Example of a function of two variables that cannot be an $R$-function |
| title_short | Example of a function of two variables that cannot be an $R$-function |
| title_sort | example of a function of two variables that cannot be an $r$-function |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2863 |
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